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Full text of "Treatise On Analysis Vol-Ii"

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184 XIII INTEGRATION then so is inf (u, 1) (same method as in (d)). Deduce that, if u and v are pure potentials, then inf(w, v + 1) is a pure potential, by remarking that v + inf(w, 1) is a pure potential and using (d). Finally show that, if fe £?% is ^>0 almost everywhere and if ue& is such that Uf(x) <j H(JC) + 1 almost everywhere w fte set of points xeX such that f(x) > 0, then \Jf(x) ^ u(x) + 1 almost everywhere in X (" complete maximum prin- ciple": same method as in (e)). 14. INTEGRATION WITH RESPECT TO A POSITIVE MEASURE WITH BASE ji (13.14.1) Let g be a locally ^.-integrable function which is ^0 on X, and let v — g • fj.. Then iffZ> 0 -is any R-valued function on X, we have (13.14.1.1) f*/Wv= t J J where, on the right-hand side, the value of fg is by definition taken to be 0 at every point xeXat which one off(x), g(x) vanishes (even if the other factor is +00 (13.11)). The proof consists of several steps. (13.14.1.2) Suppose first of all that/e</. Then (12.7.8) there exists an in- creasing sequence (fn) of functions belonging to «?TR(X) such that/= sup/n . n In view of the convention about products, this implies that the sequence (fng) is increasing and that fg is equivalent to sup(/n^), because g is almost n everywhere finite (with respect to fj). Moreover, the functions fng are ju- integrable (13.13.1), and \fngdn=.\fndv. Hence it follows from (13.5.7) that /rfv = sup /Brfv = sup /B0rf/t= fgdfi. (13.14.1.3) Every ^-negligible set N is also ^-negligible. Suppose first of all that N is relatively compact. Then ((3.18.2) and (13.7.9)) there exists a decreasing sequence of relatively compact open sets Un containing N, such that inf ju(U») = 0. Since gcpl]n is ju-integrable n by virtue of (13.13.1), we have J#<pUn dfi = v(Nn) by (13.14.1.2) and Deduce from (b) that an element v e 3f? belongs to & if and only if (v\w) i>0